Discussion

Explanation:

Consider the given diagram,

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Here, O is the center of the largest circle, A and B are the centers of the circle having radius 10 and 5 feet respectively.

Let C be the center of the largest circle that can be cut from the remaining portion.

The circles having radius 10 and 5 cm touch each other at point D.

Let radius of the largest circle that can be cut from the remaining portion be r.

Now, one can easily observe that AO = OD = BD = 5 cm.

Now, AC = 10 + r, and BC = 5 + r, and OC = 15 – r

Let DC = a.

Now, applying Apollonius in triangle ADC, we have,

(10 + r)² + a² = 2((15 – r)² + 5²)

i.e., a² – r² + 80r = 400 … (I)

Similarly, applying Apollonius theorem in triangle OCB, we get,

(15 – r)² + (5 + r)² = 2(a² + 5²)

i.e. 2a² – 2r² + 20r = 200 … (II)

By, 2 × (I) – (II), we get,

140r = 600

Hence, r = 30/7

Hence, diameter of the required circle = 60/7 ≈ 8.57

Hence, option 3.

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